In 3D, i.e. three-dimensional, beamforming systems, a beam is generated and deviated in various directions to scan a given portion of space, these beamforming systems are possibly used both for transmitting and receiving signals and may generate one beam at a time or multiple beams at the same time [1].
When the spectra of the signals to be processed comprise an excessively wide range, as it frequently occurs with passive systems, a so-called “filter-and-sum” beamforming technique is used, to optimize performances over the whole range of frequencies [2]. In these cases, in order to obtain the desired functions and avoid excessive complexity and costs, both the design of the planar or volumetric array and the FIR (Finite Impulse Response) filters must be appropriately configured.
When the range of frequencies becomes very wide, such as for sound wave processing, two distinct problems arise: on the one hand, in the lower portion of the frequency band, the wavelength is comparable to or greater than the aperture of the transducer array. This considerably reduces the directivity of the array. On the other hand, in the band portions with higher frequencies, the shorter wavelengths require a considerable number of transducers to be positioned at the aperture of the transducer array, to prevent space subsampling. Both problems may be addressed at the same time by the so-called superdirective beamforming technique at low frequencies, and by the use of sparse aperiodic transducer arrays at high frequencies.
The theory of superdirectivity is known [4-7] and involves solutions that afford increased directivity of the transducer array as compared with the directivity that can be obtained by uniform weighting. Nevertheless, the application of this theory to real systems has been long prevented by the inadequate robustness of this solution, due to the imperfections of the transducer array and to randomly fluctuating characteristics of the transducers.
Certain methods are known in the art for designing superdirective beamformers that might ensure adequate robustness in view of practical applications. Nevertheless, while the use of superdirectivity for broadband signal processing by filter-and-sum beamforming has become rather widespread [8, 9, 11-18], the application of superdirectivity to planar arrays has been considered much less frequently [10, 18].
Aperiodically arranged transducer arrays [4] may be used to prevent the occurrence of undesired lobes (known as grating lobes) when the distance between transducers is greater than half the wavelength. Various methods have been suggested in the art for synthesis of sparse aperiodic transducer arrays, with configurations obtained by analytic, stochastic or hybrid approaches, involving techniques based on simplex algorithms [19] and compressive sampling techniques [20, 21], as well as the use of genetic algorithms [22, 23], simulated annealing [24, 25] and many other optimization strategies [26-29].
In spite of what has been suggested in the art, broadband signal processing by filter-and-sum beamforming, in combination with the use of a planar transducer array having a sparse aperiodic arrangement of transducers has been rarely considered [30, 31].
Furthermore, while both the theory of superdirectivity and aperiodic configurations of transducer arrays have been known in the art, they have been basically considered as two distinct techniques.
In document [30], the configuration of planar transducer arrays is limited to two special cases, involving two-dimensional apertures, i.e. two-dimensional apertures in which transducers are placed on radial lines and two-dimensional apertures that may be separated into distributions corresponding to two one-dimensional apertures. Furthermore, this document only considers beam patterns that are invariant with respect to frequency.
On the other hand, in document [31] filter coefficient optimization and transducer placement are determined by sequential quadratic programming. Each time that a transducer is displaced, the beam pattern has to be calculated at each node of a grid that is used to discretize the three-dimensional domain of this beam pattern (frequency and two angles or lengths). Excessive increase of computational load is avoided by limiting the number of transducers and selecting a rough pitch for the discretization grid. When considering a high subsampling factor in combination with tens of transducers, the side lobes become so steep that fine discretization steps are required for accurate beam pattern estimation. Nevertheless, in this case the method of document [31] is not applicable. Furthermore, the methods of both documents [30] and [31] do not consider the synthesis of a transducer array that is required to be robust and superdirective for a significant portion of the frequency band.
Document [18] suggests a robust and superdirective planar transducer array, whose average directivity is maximized by operating both on transducer position and on FIR filter coefficients. Since directivity determination does not require full beam pattern calculation, the cost function is calculated using a discretized grid which combines the frequency axis with all the possible beam steering directions.
The method as described in document [18] cannot be practically used for configuration of a planar transducer array when the frequency range is very high and both transducer position and frequency response of the FIR filters are required to be optimized, for two reasons: at the high frequencies of the relevant range, average directivity maximization without a sidelobe control mechanism is insufficient to avoid undesired lobe or very high minor lobes; a special set of FIR filters is defined for each steering direction, which is unacceptable when considering great numbers of steering directions. This also causes a considerable increase of the computational load.
Document [3] describes a configuration method for a linear transducer array which involves contextual optimization of the position of transducers and FIR filters. The method involves the configuration of an aperiodic linear array and a broadband superdirective beamformer. Nevertheless, since optimization is carried out by minimization of a cost function that has to be recalculated multiple times on a discretization grid for discretization of the beam pattern domain, this technique cannot actually be extended to the planar array situation, due to the huge computational load required therefor. Furthermore, the broadband configuration cannot be divided into a plurality of independent problems having a narrower band, as all the narrower bands will be required to have the same optimized transducer array configuration.